Why was Cauchy careful to not say that thefundamental sequences converged into thespace from which their elements had beengiven?

I realize that you are not talking aboutthose subjects. But you are taking themto the garbage heap -- along with everyplausible piece of mathematics that usesthe completeness axiom for the real numbers.

You cannot prove the fundamental theoremof algebra without results from analysis.It requires the existence of irrationalroots for polynomials and the intermediatevalue theorem. So, you are tossingalgebra onto the same heap with analysis.

Now, there is a circularity in the topologyof real numbers. If you want to have

x=y

it must satisfy the axioms of a metricspace. But those axioms are toostrong.

Go get yourself a copy of "General Topology"by Kelley and read about uniformities andthe metrization lemma for systems of relations.

What you will find is that the metric spaceaxioms (the important direction associatedwith pseudometrics) depend on the least upperbound principle.

One can simply view it as fundamental sequencesbeing grounded by cuts. It is not circularin that sense. It simply makes Dedekind priorto Cantor.

Before you continue with this mess, you shouldtake some time to learn what it means for tworeal numbers to be equal to one another.